Problem: Solve
\[\frac{1}{x + 9} + \frac{1}{x + 7} = \frac{1}{x + 10} + \frac{1}{x + 6}.\]
Solution: Combining the fractions, we get
\[\frac{2x + 16}{(x + 9)(x + 7)} = \frac{2x + 16}{(x + 10)(x + 6)}.\]Hence,
\[(2x + 16)(x + 10)(x + 6) = (2x + 16)(x + 9)(x + 7),\]so
\[2(x + 8)[(x + 10)(x + 6) - (x + 9)(x + 7)] = 2(x + 8)(-3) = 0.\]Thus, $x = \boxed{-8}.$